498 



BRIDGE. 



Theory. Between that and the crown they must be lightened, 

 by using lighter matter, or making vacant spaces in the 

 spandrel : a:id at a greater distance from the crown, 

 the flank, although solid, will be too light for a crown 

 of 4; so that we must expand or increase the breadth 

 of the arch, in order to preserve the equilibration. 

 Every different thickness of crown will require a dif- 

 ferent arrangement in this respect. Without there- 



fore prosecuting this farther, let ua assume the thick- The 

 ness equal to -^ of the radius, or T '^ of the span '- "' 

 in a semicircle ; a proportion not unusual, and of easy 

 calculation ; from thence, to find the density of the 

 matter in the spandrel, take the numbers of line 7, 

 divide by these of line 9, and multiply by the given 

 thickness T ' 5> that is, divide by 10 ; we have for the 

 density in the spandrel, in that case, 



.26 



.'26 | .270 .289 .313 .34.6 



.394. 



.555 | .700 | .93 1 (.13) [ 



Accordingly, the density beginning at about the 

 crown, must be about - at 30 from it, and thence 

 gradually increase till about 60, where it must be. 1, 

 or equal to that of the arch. After that, if no denser 

 material can be employed, the arch must be expanded 

 in breadth, having already arrived at the limit of den- 

 sity. 



If we make the thickness at the crown -^ of radius, 

 the densities will just be -S- of the above numbers ; the 

 point of solidity will be removed a little farther from 

 the crown ; and indeed whatever be the thickness, 

 the densities will be proportional to the above num- 

 bers, and may easily be had from them. 



The above is for a horizontal roadway. There will 

 be some alteration requisite if the road be made to 

 slope up the arch : the quantity of pressure that is 

 thus lost, must be corrected by increasing the den- 

 sity of the spandrel ; and this will be more necessary 

 towards the springing. It will not be difficult for 



the practical builder to form an idea of its effect. 

 Take the section at any part, say 30 from the 

 crown, where the horizontal distance is 4- of radius; 

 suppose the road to slope 1 in 10, for example, 

 which is great, the fall will have become T '^ of ra- 

 dius, or .05 ; the versed sine is .1339, accordingly 

 the height in the spandrel is reduced to .0839, and 

 the density being increased, inversely as the height, 

 we have .552 in this case at 30 instead of .346, 

 other things being the same. Yet this density is too 

 great ; for the solid matter in the roadway will be 

 increased, being lengthened by sloping. At the 

 same time it admits of doubt, whether it may not be 

 made thinner, in the same proportion ; for its oblique 

 position gives a greater vertical thickness. This 

 will preserve the density at .552, and the whole 

 series will be found, by deducting T r - of the sine 

 from the versed sine in col. 8, and proceeding with 

 the remainders as with col. 8, as follows : 



And the effect of this height into the diff. of sines will be 



Where we find the density in no case less than ^, 

 which is about 30 from the crown, and it increases 

 both ways, about. J at 45 and at 19, and solid 

 about 53 and 16i, the first 10 are marked ne- 

 gative; for we should observe, that when we keep the 

 thickness at the crown =,'5-, the parallel to the road- 

 way cuts the curve of arch-stones. We ought in fact 

 to make the roadway of a proper thickness where 

 the arch approaches nearest to it, and relieve the 

 crown by rounding the two inclined planes into each 

 other. This will also tend to diminish the density 

 necessary in the spandrel ; for the height will be a 

 little increased, while at the same time a greater pres- 

 sure is derived from the solid roadway. But we 

 choose to allow the example to remain in this way, 

 that the reader may see that every necessary infor- 

 mation can be got, even in this way of considering 

 it. 



A smaller degree of slope, as 1 in 20, or 1 in 40, 

 will tend to diminish still farther the density neces- 

 sary in the spandrel, and . approximate it to those 

 found for the horizontal line. We might calculate 

 the densities as well for these useful slopes, at for 



other thicknesses of the crown or proportions be- 

 tween the key-stones and superincumbent roadway, 

 which, in the preceding enquiry, is taken at equality; 

 but we forbear doing so, being satisfied with giving 

 the intelligent practitioner clear ideas of the subject. 

 He already knows there are pretty wide limits to his 

 practice ; and, if the case be any way delicate, we 

 should think any person deserving the name of archi- 

 tect may, after what we have said, go over the ne- 

 cessary calculations for himself. 



The mathematical reader will perhaps say, that 

 we have taken a very awkward and unscientific mode 

 of resolving this problem ; we are not, however, in- 

 clined to admit that opinion. Our object has not 

 been to give a specimen of the application of cal- 

 culus ; but to shew the practical builder how a good 

 conception may be formed of the relative pressures 

 in different parts of his arch, and this by a process 

 purely arithmetical, and which is level to every ca- 

 pacity. We conceive that this is the way to make our 

 speculations really useful, and perhaps it were well if 

 scientific men had this oftener in view. Neither 

 have we carried our results to many figures, like 



